# Sample Extraction from RLWE to LWE

In this article I’ll derive a trick used in FHE called sample extraction. In brief, it allows one to partially convert a ciphertext in the Ring Learning With Errors (RLWE) scheme to the Learning With Errors (LWE) scheme.

Here are some other articles I’ve written about other FHE building blocks, though they are not prerequisites for this article.

## LWE and RLWE

The first two articles in the list above define the Learning With Errors problem (LWE). I will repeat the definition here:

LWE: The LWE encryption scheme has the following parameters:

• A plaintext space $\mathbb{Z}/q\mathbb{Z}$, where $q \geq 2$ is a positive integer. This is the space that the underlying message $m$ comes from.
• An LWE dimension $n \in \mathbb{N}$.
• A discrete Gaussian error distribution $D$ with a mean of zero and a fixed standard deviation.

An LWE secret key is defined as a vector $s \in \{0, 1\}^n$ (uniformly sampled). An LWE ciphertext is defined as a vector $a = (a_1, \dots, a_n)$, sampled uniformly over $(\mathbb{Z} / q\mathbb{Z})^n$, and a scalar $b = \langle a, s \rangle + m + e$, where $m$ is the message, $e$ is drawn from $D$ and all arithmetic is done modulo $q$. Note: the message $m$ usually is represented by placing an even smaller message (say, a 4-bit message) in the highest-order bits of a 32-bit unsigned integer. So then decryption corresponds to computing $b – \langle a, s \rangle = m + e$ and rounding the result to recover $m$ while discarding $e$.

Without the error term, an attacker could determine the secret key from a polynomial-sized collection of LWE ciphertexts with something like Gaussian elimination. The set of samples looks like a linear (or affine) system, where the secret key entries are the unknown variables. With an error term, the problem of solving the system is believed to be hard, and only exponential time/space algorithms are known.

RLWE: The Ring Learning With Errors (RLWE) problem is the natural analogue of LWE, where all scalars involved are replaced with polynomials over a (carefully) chosen ring.

Formally, the RLWE encryption scheme has the following parameters:

• A ring $R = \mathbb{Z}/q\mathbb{Z}$, where $q \geq 2$ is a positive integer. This is the space of coefficients of all polynomials in the scheme. I usually think of $q$ as $2^{32}$, i.e., unsigned 32-bit integers.
• A plaintext space $R[x] / (x^N + 1)$, where $N$ is a power of 2. This is the space that the underlying message $m(x)$ comes from, and it is encoded as a list of $N$ integers forming the coefficients of the polynomial.
• An RLWE dimension $n \in \mathbb{N}$.
• A discrete Gaussian error distribution $D$ with a mean of zero and a fixed standard deviation.

An RLWE secret key $s$ is defined as a list of $n$ polynomials with binary coefficients in $\mathbb{B}[x] / (x^N+1)$, where $\mathbb{B} = \{0, 1\}$. The coefficients are uniformly sampled, like in LWE. An RLWE ciphertext is defined as a vector of $n$ polynomials $a = (a_1(x), \dots, a_n(x))$, sampled uniformly over $(R[x] / (x^N+1))^n$, and a polynomial $b(x) = \langle a, s \rangle + m(x) + e(x)$, where $m(x)$ is the message (with a similar “store it in the top bits” trick as LWE), $e(x)$ is a polynomial with coefficients drawn from $D$ and all the products of the inner product are done in $R[x] / (x^N+1)$. Decryption in RLWE involves computing $b(x) – \langle a, s \rangle$ and rounding appropriately to recover $m(x)$. Just like with RLWE, the message is “hidden” in the noise added to an equation corresponding to the polynomial products (i.e., without the noise and with enough sample encryptions of the same message/secret key, you can solve the system and recover the message). For more notes on how polynomial multiplication ends up being tricker in this ring, see my negacyclic polynomial multiplication article.

The most common version of RLWE you will see in the literature sets the vector dimension $n=1$, and so the secret key $s$ is a single polynomial, the ciphertext is a single polynomial, and RLWE can be viewed as directly replacing the vector dot product in LWE with a polynomial product. However, making $n$ larger is believed to provide more security, and it can be traded off against making the polynomial degree smaller, which can be useful when tweaking parameters for performance (keeping the security level constant).

## Sample Extraction

Sample extraction is the trick of taking an RLWE encryption of $m(x) = m_0 + m_1(x) + \dots + m_{N-1}x^{N-1}$, and outputting an LWE encryption of $m_0$. In our case, the degree $N$ and the dimension $n_{\textup{RLWE}}$ of the input RLWE ciphertext scheme is fixed, but we may pick the dimension $n_{\textup{LWE}}$ of the LWE scheme as we choose to make this trick work.

This is one of those times in math when it is best to “just work it out with a pencil.” It turns out there are no serious obstacles to our goal. We start with polynomials $a = (a_1(x), \dots, a_n(x))$ and $b(x) = \langle a, s \rangle + m(x) + e(x)$, and we want to produce a vector of scalars $(x_1, \dots, x_D)$ of some dimension $D$, a corresponding secret key $s$, and a $b = \langle a, s \rangle + m_0 + e’$, where $e’$ may be different from the input error $e(x)$, but is hopefully not too much larger.

As with many of the articles in this series, we employ the so-called “phase function” to help with the analysis, which is just the partial decryption of an RLWE ciphertext without the rounding step: $\varphi(x) = b(x) – \langle a, s \rangle = m(x) + e(x)$. The idea is as follows: inspect the structure of the constant term of $\varphi(x)$, oh look, it’s an LWE encryption.

So let’s expand the constant term of $b(x) – \langle a, s \rangle$. Given a polynomial expression, I will use the notation $(-)$ to denote the constant coefficient, and $(-)[k]$ for the $k$-th coefficient.

\begin{aligned}(b(x) – \langle a, s \rangle) &= b – \left ( (a_1s_1) + \dots + (a_n s_n) \right ) \end{aligned}

Each entry in the dot product is a negacyclic polynomial product, so its constant term requires summing all the pairs of coefficients of $a_i$ and $s_i$ whose degrees sum to zero mod $N$, and flipping signs when there’s wraparound. In particular, a single product above for $a_i s_i$ has the form:

$$(a_is_i)  = s_ia_i – s_ia_i[N-1] – s_ia_i[N-2] – \dots – s_i[N-1]a_i$$

Notice that I wrote the coefficients of $s_i$ in increasing order. This was on purpose, because if we re-write this expression $(a_is_i)$ as a dot product, we get

$$(a_is_i) = \left \langle (s_i, s_i, \dots, s_i[N-1]), (a_i, -a_i[N-1], \dots, -a_i)\right \rangle$$

In particular, the $a_i[k]$ are public, so we can sign-flip and reorder them easily in our conversion trick. But $s_i$ is unknown at the time the sample extraction needs to occur, so it helps if we can leave the secret key untouched. And indeed, when we apply the above expansion to all of the terms in the computation of $\varphi(x)$, we end up manipulating the $a_i$’s a lot, but merely “flattening” the coefficients of $s = (s_1(x), \dots, s_n(x))$ into a single long vector.

So combining all of the above products, we see that $(b(x) – \langle a, s \rangle)$ is already an LWE encryption with $(x, y) = ((x_1, \dots, x_D), b)$, and $x$ being the very long ($D = n*N$) vector

\begin{aligned} x = (& a_0, -a_0[N-1], \dots, -a_0, \\ &a_1, -a_1[N-1], \dots, -a_1, \\ &\dots , \\ &a_n, -a_n[N-1], \dots, -a_n ) \end{aligned}

And the corresponding secret key is

\begin{aligned} s_{\textup{LWE}} = (& (s_0, s_0, \dots, s_0[N-1] \\ &(s_1, s_1, \dots, s_1[N-1], \\ &\dots , \\ &s_n, s_n, \dots, s_n[N-1] ) \end{aligned}

And the error in this ciphertext is exactly the constant coefficient of the error polynomial $e(x)$ from the RLWE encryption, which is independent of the error of all the other coefficients.

## Commentary

This trick is a best case scenario. Unlike with key switching, we don’t need to encrypt the output LWE secret key to perform the conversion. And unlike modulus switching, there is no impact on the error growth in the conversion from RLWE to LWE. So in a sense, this trick is “perfect,” though it loses information about the other coefficients of $m(x)$ in the process. As it happens, the CGGI FHE scheme that these articles are building toward only uses the constant coefficient.

The only twist to think about is that the output LWE ciphertext is dependent on the RLWE scheme parameters. What if you wanted to get a smaller-dimensional LWE ciphertext as output? This is a realistic concern, as in the CGGI FHE scheme one starts from an LWE ciphertext of one dimension, goes to RLWE of another (larger) dimension, and needs to get back to LWE of the original dimension by the end.

To do this, you have two options: one is to pick the RLWE ciphertext parameters $n, N$, so that their product is the value you need. A second is to allow the RLWE parameters to be whatever you need for performance/security, and then employ a key switching operation after the sample extraction to get back to the LWE parameters you need.

It is worth mentioning—though I am far from fully understanding the methods—there other ways to convert between LWE and RLWE. One can go from LWE to RLWE, or from a collection of LWEs to RLWE. Some methods can be found in this paper and its references.

Until next time!

# Estimating the Security of Ring Learning with Errors (RLWE)

This article was written by my colleague, Cathie Yun. Cathie is an applied cryptographer and security engineer, currently working with me to make fully homomorphic encryption a reality at Google. She’s also done a lot of cool stuff with zero knowledge proofs.

In previous articles, we’ve discussed techniques used in Fully Homomorphic Encryption (FHE) schemes. The basis for many FHE schemes, as well as other privacy-preserving protocols, is the Learning With Errors (LWE) problem. In this article, we’ll talk about how to estimate the security of lattice-based schemes that rely on the hardness of LWE, as well as its widely used variant, Ring LWE (RLWE).

A previous article on modulus switching introduced LWE encryption, but as a refresher:

## Reminder of LWE

A literal repetition from the modulus switching article. The LWE encryption scheme I’ll use has the following parameters:

• A plaintext space $\mathbb{Z}/q\mathbb{Z}$, where $q \geq 2$ is a positive integer. This is the space that the underlying message comes from.
• An LWE dimension $n \in \mathbb{N}$.
• A discrete Gaussian error distribution $D$ with a mean of zero and a fixed standard deviation.

An LWE secret key is defined as a vector in $\{0, 1\}^n$ (uniformly sampled). An LWE ciphertext is defined as a vector $a = (a_1, \dots, a_n)$, sampled uniformly over $(\mathbb{Z} / q\mathbb{Z})^n$, and a scalar $b = \langle a, s \rangle + m + e$, where $e$ is drawn from $D$ and all arithmetic is done modulo $q$. Note that $e$ must be small for the encryption to be valid.

## Learning With Errors (LWE) security

Choosing appropriate LWE parameters is a nontrivial challenge when designing and implementing LWE based schemes, because there are conflicting requirements of security, correctness, and performance. Some of the parameters that can be manipulated are the LWE dimension $n$, error distribution $D$ (referred to in the next few sections as $X_e$), secret distribution $X_s$, and plaintext modulus $q$.

## Lattice Estimator

Here is where the Lattice Estimator tool comes to our assistance! The lattice estimator is a Sage module written by a group of lattice cryptography researchers which estimates the concrete security of Learning with Errors (LWE) instances.

For a given set of LWE parameters, the Lattice Estimator calculates the cost of all known efficient lattice attacks – for example, the Primal, Dual, and Coded-BKW attacks. It returns the estimated number of “rops” or “ring operations” required to carry out each attack; the attack that is the most efficient is the one that determines the security parameter. The bits of security for the parameter set can be calculated as $\log_2(\text{rops})$ for the most efficient attack.

## Running the Lattice Estimator

For example, let’s estimate the security of the security parameters originally published for the popular TFHE scheme:

n = 630
q = 2^32
Xs = UniformMod(2)
Xe = DiscreteGaussian(stddev=2^17)


After installing the Lattice Estimator and sage, we run the following commands in sage:

> from estimator import *
> schemes.TFHE630
LWEParameters(n=630, q=4294967296, Xs=D(σ=0.50, μ=-0.50), Xe=D(σ=131072.00), m=+Infinity, tag='TFHE630')
> _ = LWE.estimate(schemes.TFHE630)
bkw                  :: rop: ≈2^153.1, m: ≈2^139.4, mem: ≈2^132.6, b: 4, t1: 0, t2: 24, ℓ: 3, #cod: 552, #top: 0, #test: 78, tag: coded-bkw
usvp                 :: rop: ≈2^124.5, red: ≈2^124.5, δ: 1.004497, β: 335, d: 1123, tag: usvp
bdd                  :: rop: ≈2^131.0, red: ≈2^115.1, svp: ≈2^131.0, β: 301, η: 393, d: 1095, tag: bdd
bdd_hybrid           :: rop: ≈2^185.3, red: ≈2^115.9, svp: ≈2^185.3, β: 301, η: 588, ζ: 0, |S|: 1, d: 1704, prob: 1, ↻: 1, tag: hybrid
bdd_mitm_hybrid      :: rop: ≈2^265.5, red: ≈2^264.5, svp: ≈2^264.5, β: 301, η: 2, ζ: 215, |S|: ≈2^189.2, d: 1489, prob: ≈2^-146.6, ↻: ≈2^148.8, tag: hybrid
dual                 :: rop: ≈2^128.7, mem: ≈2^72.0, m: 551, β: 346, d: 1181, ↻: 1, tag: dual
dual_hybrid          :: rop: ≈2^119.8, mem: ≈2^115.5, m: 516, β: 314, d: 1096, ↻: 1, ζ: 50, tag: dual_hybrid


In this example, the most efficient attack is the dual_hybrid attack. It uses 2^119.8 ring operations, and so these parameters provide 119.8 bits of security. The reader may notice that the TFHE website claims those parameters give 128 bits of security. This discrepancy is due to the fact that they used an older library (the LWE estimator, which is no longer maintained), which doesn’t take into account the most up-to-date lattice attacks.

For further reading, Benjamin Curtis wrote an article about parameter selection for the CONCRETE implementation of the TFHE scheme. Benjamin Curtis, Martin Albrecht, and other researchers also used the Lattice Estimator to estimate all the LWE and NTRU schemes.

## Ring Learning with Errors (RLWE) security

It is often desirable to use Ring LWE instead of LWE, for greater efficiency and smaller key sizes (as Chris Peikert illustrates via meme). We’d like to estimate the security of a Ring LWE scheme, but it wasn’t immediately obvious to us how to do this, since the Lattice Estimator only operates over LWE instances. In order to use the Lattice Estimator for this security estimate, we first needed to do a reduction from the RLWE instance to an LWE instance.

## Attempted RLWE to LWE reduction

Given an RLWE instance with $\text{RLWE_dimension} = k$ and $\text{poly_log_degree} = N$, we can create a relation that looks like an LWE instance of $\text{LWE_dimension} = N * k$ with the same security, as long as $N$ is a power of 2 and there are no known attacks that target the ring structure of RLWE that are more efficient than the best LWE attacks. Note: $N$ must be a power of 2 so that $x^N+1$ is a cyclotomic polynomial.

An RLWE encryption has the following form: $(a_0(x), a_1(x), … a_{k-1}(x), b(x))$

•   Public polynomials: $a_0(x), a_1(x), \dots a_{k-1}(x) \overset{{\scriptscriptstyle\$}}{\leftarrow} (\mathbb{Z}/{q \mathbb{Z}[x]} ) / (x^N + 1)^k$• Secret (binary) polynomials:$ s_0(x), s_1(x), \dots s_{k-1}(x) \overset{{\scriptscriptstyle\$}}{\leftarrow} (\mathbb{B}_N[x])^k$
•   Error: $e(x) \overset{{\scriptscriptstyle\$}}{\leftarrow} \chi_e$• RLWE instance:$ b(x) = \sum_{i=0}^{k-1} a_i(x) \cdot s_i(x) + e(x) \in (\mathbb{Z}/{q \mathbb{Z}[x]} ) / (x^N + 1)$We would like to express this in the form of an LWE encryption. We can make start with the simple case, where$ k=1 $. Therefore, we will only be working with the zero-entry polynomials,$a_0(x)$and$s_0(x)$. (For simplicity, in the next example you can ignore the zero-subscript and think of them as$a(x)$and$s(x)$). ## Naive reduction for$k=1$(wrong!) Naively, if we simply defined the LWE$A$matrix to be a concatenation of the coefficients of the RLWE polynomial$a(x)$, we get: $$A_{\text{LWE}} = ( a_{0, 0}, a_{0, 1}, \dots a_{0, N-1} )$$ We can do the same for the LWE$s$vector: $$s_{\text{LWE}} = ( s_{0, 0}, s_{0, 1}, \dots s_{0, N-1} )$$ But this doesn’t give us the value of$b_{LWE}$for the LWE encryption that we want. In particular, the first entry of$b_{LWE}$, which we can call$b_{\text{LWE}, 0}$, is simply a product of the first entries of$a_0(x)$and$s_0(x)$: $$b_{\text{LWE}, 0} = a_{0, 0} \cdot s_{0, 0} + e_0$$ However, we want$b_{\text{LWE}, 0}$to be a sum of the products of all the coefficients of$a_0(x)$and$s_0(x)$that give us a zero-degree coefficient mod$x^N + 1$. This modulus is important because it causes the product of high-degree monomials to “wrap around” to smaller degree monomials because of the negacyclic property, such that$x^N \equiv -1 \mod x^N + 1$. So the constant term$b_{\text{LWE}, 0}should include all of the following terms: \begin{aligned} b_{\text{LWE}, 0} = & a_{0, 0} \cdot s_{0, 0} \\ – & a_{0, 1} \cdot s_{0, N-1} \\ – & a_{0, 2} \cdot s_{0, N-2} \\ – & \dots \\ – & a_{0, N-1} \cdot s_{0, 1}\\ + & e_0\\ \end{aligned} ## Improved reduction fork=1$We can achieve the desired value of$b_{\text{LWE}}$by more strategically forming a matrix$A_{\text{LWE}}$, to reflect the negacyclic property of our polynomials in the RLWE space. We can keep the naive construction for$s_\text{LWE}$. $$A_{\text{LWE}} = \begin{pmatrix} a_{0, 0} & -a_{0, N-1} & -a_{0, N-2} & \dots & -a_{0, 1}\\ a_{0, 1} & a_{0, 0} & -a_{0, N-1} & \dots & -a_{0, 2}\\ \vdots & \ddots & & & \vdots \\ a_{0, N-1} & \dots & & & a_{0, 0} \\ \end{pmatrix}$$ This definition of$A_\text{LWE}$gives us the desired value for$b_\text{LWE}$, when$b_{\text{LWE}}$is interpreted as the coefficients of a polynomial. As an example, we can write out the elements of the first row of$b_\text{LWE}: \begin{aligned} b_{\text{LWE}, 0} = & \sum_{i=0}^{N-1} A_{\text{LWE}, 0, i} \cdot s_{0, i} + e_0 \\ b_{\text{LWE}, 0} = & a_{0, 0} \cdot s_{0, 0} \\ – & a_{0, 1} \cdot s_{0, N-1} \\ – & a_{0, 2} \cdot s_{0, N-2} \\ – & \dots \\ – & a_{0, N-1} \cdot s_{0, 1}\\ + & e_0 \\ \end{aligned} ## Generalizing for allk$In the generalized$k$case, we have the RLWE equation: $$b(x) = a_0(x) \cdot s_0(x) + a_1(x) \cdot s_1(x) \cdot a_{k-1}(x) \cdot s_{k-1}(x) + e(x)$$ We can construct the LWE elements as follows: $$A_{\text{LWE}} = \left ( \begin{array}{c|c|c|c} A_{0, \text{LWE}} & A_{1, \text{LWE}} & \dots & A_{k-1, \text{LWE}} \end{array} \right )$$ where each sub-matrix is the construction from the previous section: $$A_{\text{LWE}} = \begin{pmatrix} a_{i, 0} & -a_{i, N-1} & -a_{i, N-2} & \dots & -a_{i, 1}\\ a_{i, 1} & a_{i, 0} & -a_{i, N-1} & \dots & -a_{i, 2}\\ \vdots & \ddots & & & \vdots \\ a_{i, N-1} & \dots & & & a_{i, 0} \\ \end{pmatrix}$$ And the secret keys are stacked similarly: $$s_{\text{LWE}} = ( s_{0, 0}, s_{0, 1}, \dots s_{0, N-1} \mid s_{1, 0}, s_{1, 1}, \dots s_{1, N-1} \mid \dots )$$ This is how we can reduce an RLWE instance with RLWE dimension$k$and polynomial modulus degree$N$, to a relation that looks like an LWE instance of LWE dimension$N * k$. ## Caveats and open research This reduction does not result in a correctly formed LWE instance, since an LWE instance would have a matrix$A$that is randomly sampled, whereas the reduction results in an matrix$A$that has cyclic structure, due to the cyclic property of the RLWE instance. This is why I’ve been emphasizing that the reduction produces an instance that looks like LWE. All currently known attacks on RLWE do not take advantage of the structure, but rather directly attack this transformed LWE instance. Whether the additional ring structure can be exploited in the design of more efficient attacks remains an open question in the lattice cryptography research community. In her PhD thesis, Rachel Player mentions the RLWE to LWE security reduction: In order to try to pick parameters in Ring-LWE-based schemes (FHE or otherwise) that we hope are sufficiently secure, we can choose parameters such that the underlying Ring-LWE instance should be hard to solve according to known attacks. Each Ring-LWE sample can be used to extract$n$LWE samples. To the best of our knowledge, the most powerful attacks against$d$-sample Ring-LWE all work by instead attacking the$nd$-sample LWE problem. When estimating the security of a particular set of Ring-LWE parameters we therefore estimate the security of the induced set of LWE parameters. This indicates that we can do this reduction for certain RLWE instances. However, we must be careful to ensure that the polynomial modulus degree$N$is a power of two, because otherwise the error distribution “breaks”, as my colleague Baiyu Li explained to me in conversation: The RLWE problem is typically defined in using the ring of integers of the cyclotomic field$\mathbb{Q}[X]/(f(X))$, where$f(X)$is a cyclotomic polynomial of degree$k=\phi(N)$(where$\phi$is Euler’s totient function), and the error is a spherical Gaussian over the image of the canonical embedding into the complex numbers$\mathbb{C}^k$(basically the images of primitive roots of unity under$f$). In many cases we set$N$to be a power of 2, thus$f(X)=X^{N/2}+1$, since the canonical embedding for such$N$has a nice property that the preimage of the spherical Gaussian error is also a spherical Gaussian over the coefficients of polynomials in$\mathbb{Q}[X]/(f(X))$. So in this case we can sample$k=N/2$independent Gaussian numbers and use them as the coefficients of the error polynomial$e(x)$. For$N$not a power of 2,$f(X)$may have some low degree terms, and in order to get the spherical Gaussian with the same variance$s^2$in the canonical embedding, we probably need to use a larger variance when sampling the error polynomial coefficients. The RLWE we frequently use in practice is actually a specialized version called “polynomial LWE”, and instantiated with$N$= power of 2 and so$f(X)=X^{N/2}+1$. For other parameters the two are not exactly the same. This paper has some explanations: https://eprint.iacr.org/2018/170.pdf The error distribution “breaks” if$N$is not a power of 2 due to the fact that the precise form of RLWE is not defined on integer polynomial rings$R = \mathbb{Z}[X]/(f(X))$, but is defined on its dual (or the dual in the underlying number field, which is a fractional ideal of$\mathbb{Q}[X]/(f(x))$), and the noise distribution is on the Minkowski embedding of this dual ring. For non-power of 2$N$, the product mod$f$of two small polynomials in$\mathbb{Q}[X]/(f(x))$may be large, where small/large means their L2 norm on the coefficient vector. This means that in order to sample the required noise distribution, you may need a skewed coefficient distribution. Only when$N$is a power of 2, the dual of$R$is a scaling of$R$, and distance in the embedding of$R^{\text{dual}}$is preserved in$R$, and so we can just sample iid gaussian coefficient to get the required noise. Because working with a power-of-two RLWE polynomial modulus gives “nice” error behavior, this parameter choice is often recommended and chosen for concrete instantiations of RLWE. For example, the Homomorphic Encryption Standard recommends and only analyzes the security of parameters for power-of-two cyclotomic fields for use in homomorphic encryption (though future versions of the standard aim to extend the security analysis to generic cyclotomic rings): We stress that when the error is chosen from sufficiently wide and “well spread” distributions that match the ring at hand, we do not have meaningful attacks on RLWE that are better than LWE attacks, regardless of the ring. For power-of-two cyclotomics, it is sufficient to sample the noise in the polynomial basis, namely choosing the coefficients of the error polynomial$e \in \mathbb{Z}[x] / \phi_k(x)\$ independently at random from a very “narrow” distribution.

Existing works analyzing and targeting the ring structure of RLWE include:

It would of course be great to have a definitive answer on whether we can be confident using this RLWE to LWE reduction to estimate the security of RLWE based schemes. In the meantime, we have seen many Fully Homomorphic Encryption (FHE) schemes using this RLWE to LWE reduction, and we hope that this article helps explain how that reduction works and the existing open questions around this approach.